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I recently read Palle Yourgrau's book "A World Without Time" about Gödel's contribution to the nature of time in general relativity.

Gödel published his discovery of closed timelike curves in 1949. Many years later (in 1961), S. Chandrasekhar and James P. Wright pointed out in "The geodesic in Gödel's universe" that these curves are not geodesics, and hence Gödel's philosophical conclusions might be questionable. Again some years later, the philosopher Howard Stein pointed out that Gödel never claimed that these curves are geodesics, which Gödel confirmed immediately. Again much later other physicists have computed that these closed timelike curve must be so strongly accelerated that the energy for a particle with a finite rest mass required to run through such a curve is many times its rest mass. (I admit that I may have misunderstood/misinterpreted this last part.)


  1. This makes me wonder whether any particle (with finite rest mass) actually traveling on a closed timelike curve wouldn't violate the conservation of energy principle. (As pointed out in the comments, I made a hidden assumption here. I implicitly assumed that the particle traverses the closed timelike curve not only once or only a finite number of times, but "forever". I put "forever" in quotes, because the meaning of "forever" seems to depend on the notion of time.)
  2. I vaguely remember that light will always travel on a geodesic. Is this correct? Is this a special case of a principle that any particle in the absence of external forces (excluding gravity) will always travel on a geodesic?
  3. Is it possible for a particle to be susceptible to external forces and still have zero rest mass?
  4. Is it possible that Chandrasekhar and Wright were actually right in suggesting that Gödel's philosophical conclusions are questionable, and that they hit the nail on the head by focusing on the geodesics in the Gödel's universe?
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This makes me wonder whether any particle (with finite rest mass) actually traveling on a closed timelike curve wouldn't violate the conservation of energy principle. I don't think this follows at all from the fact that you need to input $E > mc^2$ to get a particle of mass $m$ around the CTC. You just need an external source of energy. – Ben Crowell Aug 25 '13 at 22:16
@BenCrowell Regarding your comment, I had made the implicit assumption that a particle which traverses a closed timelike curve once is forced to do so forever. If it is possible for a particle to traverse a closed timelike curve only a finite number of times, then I agree that there is no need to worry about the conservation of energy principle. – Thomas Klimpel Aug 25 '13 at 23:49
@ThomasKlimpell: CTC's are not elementary ideas. I have added a very crude analogy to give you a sense of the 'unphysical-ness' of the concept. Quite often in theoretical physics, ideas cannot be fully visualized with perfect detail. I urge you to read up the theory of causal structure. – dj_mummy Aug 26 '13 at 7:11

2 Answers 2

up vote 3 down vote accepted

I have not really studied Godel's metric, so I will only address questions 2 and 3 in a general metric (without specifically referring to Godel's metric).

Yes, light (in vacuum) will always travel on a null geodesics. Yes, particles remain on geodesics in absence of a net external force. Momentum means different things in the massive and mass-less case, since massive particles move on geodesics with timelike tangent vectors and mass-less move on null tangent vectors. 4-force is equal to the covariant derivative of 4-momentum along the tangent vector to its worldline. I will elaborate:

Let us assume a world in which quantum mechanics is bogus and all particles have a 'kick' (momentum) associated with them. A particle of light has a definite momentum associated with it. So its 'kick' can be redirected and/or diminished. Particles with mass also have this 'kick' and can also have it redirected and/or diminished. The 'kick' is redirected when 'kick' makes contact with the force applier i.e. they would be deviated from their geodesic motion.

Now, in particles with non-zero mass this kick is directly proportional to the 4-velocity. So applying a force on the particles changes its 4-velocity and deviates it from timelike geodesic motion.

However, for mass-less particles the 4-velocity does not exist (as proper time in their frame is 0). Applying force on the particle would also deviate it from null geodesic motion, but the tangent vector of it's motion would remain a null vector, so their net speed would still remain c in your local frame throughout the application of force.

Back to reality. In classical GR, we don't have any forces for these mass-less particles, but have forces (Electromagnetic forces) for massive particles. So we treat mass-less particles purely as waves with energy and momentum (that can't be changed by applying classical force). Note, in classical GR, in vacuum the speed of the EM wave can be reduced in dielectric media, but the fastest speed possible in the dielectric frame will still remain a null vector (speed of light).

In the above discussion, I treat gravitation as the structure of space-time and not as a force.

In Quantum Field Theory, observation is discontinuous and particles change in number and type between 2 successive measurements. There is a symmetry in these changes which leave net Energy and momentum invariant. So here force is irrelevant here and we treat photons as particles again.

Questions 1 and 4:

First of all there is no global conservation of energy in general relativity. There is only local conservation of energy. There are other methods used to get globally conserved quantities (like Killing vectors fields).

CTC's are looked upon as pathological entities. A whole lot of concepts in classical GR have to be revisited if if we accept CTC's in the acceptable causal structure of realistic spacetimes.

A lot of ideas we take for granted are thrown to the winds in such extreme spacetime. Let me give you a very crude and rough analogy:

-There is an astral chicken that lays an egg and dies, the egg hatches and the chick eats the egg and its parent, lays an egg, dies and so on..... Thus, the astral chicken's wordline is a CTC.

-Let's say you (moving along a normal geodesic) are at the event P (hatching of egg) and stay with the chicken till event Q (dying of chicken), the chicken will vanish suddenly after Q. Can you imagine the chicken vanishing?

-The egg also appeared suddenly in your past at P. Kind of like Marty in Back To the Future who appears and disappears suddenly. The egg-mass appears, turns into a chick and disappears, obviously from your viewpoint, energy is not conserved at all not even locally.

This is the best I can do without referring using math. Causal Structure is a very elementary theory, you will be able to understand it. This would help you better understand CTC's, which are not elementary at all. I recommend Wald's book on GR. In addition, here is a pdf by Thorne on some implications of CTCs. It is a moderately advanced paper, but very interesting.

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I'm confused by the "No" in "Yes, light will always travel on a geodesic. No, in classical GR, force only implies change in momentum." I initially interpreted it as the answer to the second part of question 2: "Is this a special case of a principle that any particle in the absence of external forces (excluding gravity) will always travel on a geodesic?" But because this answer would be unexpected for me, I googled and found a wikipedia page explaining Geodesics in general relativity. So what does the "No" really mean? – Thomas Klimpel Aug 26 '13 at 7:47
I'm more reminded of "Ouroborus" in Red Dwarf than Marty in Back to the Future by your chicken and egg. – WetSavannaAnimal aka Rod Vance Aug 26 '13 at 7:50
@ThomasKlimpel My answer had a lot of ambiguities. I edited it for greater clarity. I hope this version will avoid confusion. – dj_mummy Aug 26 '13 at 13:55
@WetSavannaAnimalakaRodVance But Ouroborus is continuous. There were discontinuities in Marty's exit and entry at different events as far as onlookers were concerned. In the chicken's frame, Ouroborus would be more apt, but not in the frame of onlookers. – dj_mummy Aug 26 '13 at 14:03

Light will not travel on a geodesic in the background spacetime, as light is an electromagnetic field and has stress-energy and so itself warps the spacetime around it. However if the light induced curvature is small compared to the curvature already there, then it won't change things much, so will go on an approximately geodesic route.

Same for you spaceship and all it's fuel. But again, as long as the ship induced curvature is not large compare to the curvature without the ship the changes are not large. And again, the comparison is ship-induced curvature compare to the curvature without the ship, it has nothing to do with the fueled up energy versus empty ship energy, well as long as the fueled up ship is the one that doesn't change the curvature much.

OK, let's get to that fuel required. Imagine two ships. The Space Valdez and the Space Chicken. One time in Camp Music, in the past, the Space Valdez is manufactured at Music Camp. It harvests fuel at Music Camp, way more fuel than it personally needs and plans to transport some fuel to Camp Parts Unknown, which is far away. So far away that the Space Valdez carries factories capable of manufacturing and assembling replacement spaceship parts. The occupants of Music Camp wish to be good galactic occupants so instruct the Space Valdez to respond with overwhelming kindness to distress broadcasts even if it uses up much fuel and resources. Who knows maybe there is a default way to program fuel transports to react so that others can recognize that it isn't a giant missile and maybe making it respond to any request to fuel or manufacture is a way to do that. Not saying that it is smart, but say it happened, even just once, in the history of the universe. So the Space Valdez sets off on its maiden voyage, fueled up, with manufacturing and communications capability and full of goodwill. Aims towards Camp Parts Unknown and goes.

So the Space Valdez is transporting a huge amount of fuel taking a nice geodesic through spacetime when it encounters a distress broadcast from a decrepit and low fuel ship called the Space Chicken. The Space Valdez accelerates to join and matches the accelerating trajectory of the Space Chicken and exchanges vast data back and forth at extremely close range. This encounter is considered productive and the Space Valdez uploads a datastore from the Space Chicken that describes an earlier state of the Space Chicken in a surprising similar looking region of spacetime. It repairs, refuels and wholesale rebuilds the Space Chicken as needed. Including resetting the whole datastore of the Space Chicken. Then they bid farewell and the Space Valdez heads back out to Camp Unknown.

Meanwhile the rejuvinated Space Chicken is in bootup mode and runs on autopilot mode which includes accelerating hard to test activate the rockets. It saves a checkpoint in the datastore (that includes instructions about how to build and repair the Space Chicken) as part of bootup process. It then uses the newly acquired fuel and polished craft to hurtle around the cylinder on an accelerated trajectory. And it made the log of it's current state shortly after losing communication distance with the Space Valdez. Eventually battered and low on fuel the Space Chicken, so very damaged but having always maintained a good record (at cost to degredation of the rest of the ship and using up of fuel) of it's earlier state and keeping the communication system good and full of data of the kind that strangers might find beneficial to have, the Space Chicken starts to broadcast a distress broadcast. This transmission is picked up by the recently manufactured Space Valdez on its first voyage.

OK, so we have the complete picture. All the fuel that is shot out of the Space Chicken's engines can be traced back as having come from the Space Valdez and then originally from Camp Music. Same with all the wear and tear energy that inevitable happened as the Space Valdez fixed up the Space Chicken. The information about how to manufacture, repair, build, program, and reboot the Space Chicken are a type of Djinn, with no ultimate source, but merely flow from one region of spacetime to another. In every local region, the physics is normal. Parts wear out, systems are used to maintain redundancy, material is used to protect the integrity of data, energy and momentum are shot out one way to accelerate the other way sop fuels goes down except when being refueled in flight by a fuel transport vessel. Nothing weird at all. Particularly no violations of conservation of energy, because that is a local law so we just need small coordinate patches and in each patch the energy changes by exactly how much flowed in. This scenario solves that fine. We could write a chapter book about each region of spacetime over an small bit of space and a small bit of time and the story looks perfectly ordinary.

As for entropy. In a sense the same thing happens, at each point entropy is already there and as time goes on the local entropy goes down by an amount less than the entropy that flows out. The Space Chicken uses fuel to do things and emits waste heat during the whole trip. With local stores of entropy, it is again a local law.

With no quantum, there is no problem. And no violation of laws of energy or thermodynamics. Other answers that mention things vanishing and appearing are wrong. General Relativity is a local theory, just break it up into small pieces where nothing weird happens, if you can't do that then you've done it wrong. If when you glue it all together you look at it and you don't like it, that's your problem if each local piece followed the rules just fine. And you don't ever even need to bring up words like forever. Just break it into pieces and describe each piece in a totally normal way and make it clear how the pieces line up.

I hope I did that for you in this example, and if you break it into pieces there is never a need for the word forever, and breaking it into pieces each of which looks fine, and sewing them together is what General Relativity is all about.

For your other questions:

Is it possible for a particle to be susceptible to external forces and still have zero rest mass?

A zero mass particle can still have its momentum changed by an external agent/interaction, and that can make it deviate from a geodesic, but unlike for a massive particle it need not, it could just increase its energy without changing direction so could continue the same direction.

Is it possible that Chandrasekhar and Wright were actually right in suggesting that Gödel's philosophical conclusions are questionable, and that they hit the nail on the head by focusing on the geodesics in the Gödel's universe?

The focus on geodesics is a red herring and a wild goose chase. Nothing travels on a geodesic unless it has no energy, no stress, no momentum and no spin. Plus you can refuel so there is no issue whatsoever. There are problems with Closed Timelike Curves, but they are more about quantum effects or stability issues.

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Good points. However, I'm not sure whether the fact that the required energy is many times the rest mass will make it necessary to recharge more than once to even complete a single cycle on the curve. But you are right, even if it is required to recharge 100 times, then Space Chicken simply needs to be lucky enough to meet 100 friendly Space Valdez vessels. – Thomas Klimpel Jun 15 at 7:48

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